Nash equilibrium in mixed strategies: penaly kicks in football

Abstract

Penalty kicks in football form a classic case of mixed strategy. The striker and the goal-keeper should, thus, have a certain probability of kicking or diving in each direction for every penalty. In this project, we examine a database of penalty kicks taken in the Premier League, the Spanish Premiera Liga and the German Bundesliga across the last 2 seasons from 2012-14. Using this database, we attempt to re-test the hypotheses in the original paper on the subject by Levitt et al (2002).

The soccer penalty process is modeled such that each kicker can take one of three possible actions {left, center, right}and each keeper can similarly dive in one of the three directions. General assumptions regarding the structure of the payoff (i.e., scoring probabilities) matrix are made. Ideally the payoff matrix is match and player -specific and thus while the theoretical predictions hold for a particular matrix, they may not be robust to aggregation. Several implications of the model are preserved by aggregation; however other basic predictions do not survive aggregation in the presence of heterogeneity.

Our analysis has been broken into 3 parts. Initially, we examine a penalty kick to see if it is a simultaneous move game. Only if that is true can we conclude or even begin to examine if mixed strategies are being applied. The second part looks at some general observations from the data that will help empirically prove several hypotheses. Finally, we test for mixed strategies under the assumption of “identical goal –keepers”. This assumption allows us to draw some powerful conclusions from the data.